Finite Group Databasebeta

← #74880

Subgroups — #74880

Order 384.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#937C211yesmaximal, characteristic
2C2×C4×#67C212yesmaximal, complemented
3C2×D3×C22⋊C4C212yesmaximal, complemented
4C3×#386C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#4956C212yesmaximal, complemented
7#4945C212yesmaximal, complemented
8#4668C212yesmaximal, complemented
9#4666C212yesmaximal, complemented
10#2358—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×C3×D4C2211yescharacteristic
13C23×C3×C4C2211yescharacteristic, complemented
14C22×#67C2212yes—
15C22×C4×D3C2212yes—
16C22×C4×D3C2212yes—
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic, complemented
19C2×#217C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2212yescomplemented
22#937C418yescomplemented
23C4×#67—28no—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic, complemented
28C2×Dic3⋊C4C2212yescomplemented
29C2×C4×Dic3C2212yescomplemented
30C2×C4×Dic3C2211yescharacteristic, complemented
31D3×C22⋊C4—28no—
32C3×#95C2212yescomplemented
33C3⋊#95C2212yescomplemented
34C3⋊#95C2212yescomplemented
35C2×#120—31no—
36C2×C4×D4—32no—
37C22×C22⋊C4—32no—
38#390—31no—
39#388—32no—
40#386—32no—
41#386D311yescharacteristic, complemented
42#385—32no—
43C4×C22⋊C4—32no—
44C24×C3C2311yescharacteristic
45C24×C3C2311yescharacteristic
46C23×D3—24no—
47C23×D3—24no—
48C23×D3—41no—
49C23×D3—41no—
50C23×D3C2×C414yescomplemented
51C23×D3C2312yes—
52C2×C3×D4—41no—
53C2×C3×D4—41no—
54C2×C3×D4C2×C414yescomplemented
55C22×C3×C4—24no—
56C22×C3×C4—41no—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2312yes—
60C2×#67—24no—
61C2×#67—24no—
62C2×#67C2×C418yescomplemented
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C2×C4×D3—24no—
67C2×C4×D3—24no—
68C2×C4×D3—24no—
69C2×C4×D3—24no—
70C2×C4×D3—42no—
71C2×C4×D3—42no—
72C3×C22⋊C4—24no—
73C3×C22⋊C4—41no—
74#217—24no—
75#217—24no—
76#217C2×C414yescomplemented
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4C2×C418yescomplemented
79(C2×D3)⋊C4D418yescomplemented
80C4⋊Dic3—41no—
81Dic3⋊C4—24no—
82C4×Dic3—24no—
83C4×Dic3—41no—
84C25—31no—
85C22×D4—32no—
86C22×D4C2×D311yescharacteristic
87C23×C4—32no—
88C23×C4—32no—
89C23×C4C2×D311yescharacteristic, complemented
90#120—38no—
91C4×D4—68no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—31no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—68no—
101C2×C22⋊C4C2×D312yes—
102C2×C22⋊C4C2×D311yescharacteristic, complemented
103C2×C42—32no—
104C2×C42—31no—
105#95—32no—
106#95—32no—
107#95C2×D312yescomplemented
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3C22×C412yes—
117C23×C3C22×C412yes—
118C23×C3C2411yescharacteristic
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—216no—
124C22×D3—44no—
125C22×D3—44no—
126C22×D3—48no—
127C22×D3—44no—
128C22×D3C22×C414yes—
129C22×D3C2×D414yes—
130C22×D3#4114yes—
131C3×D4—44no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—44no—
140C2×C3×C4C22×C412yes—
141C2×C3×C4C2×D414yes—
142#67—216no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3C22×C414yes—
151C2×Dic3C2×D414yes—
152C4×D3—48no—
153C4×D3—48no—
154C24—32no—
155C24—34no—
156C24—64no—
157C24—64no—
158C24—121no—
159C24—121no—
160C24C22×D311yescharacteristic
161C24C22×D311yescharacteristic
162C2×D4—38no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—64no—
166C2×D4—64no—
167C2×D4C4×D314yescomplemented
168C22×C4—24no—
169C22×C4—32no—
170C22×C4—32no—
171C22×C4—32no—
172C22×C4—41no—
173C22×C4—64no—
174C22×C4—64no—
175C22×C4—64no—
176C22×C4—64no—
177C22×C4—122no—
178C22×C4—122no—
179C22×C4C22×D312yes—
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D312yes—
182C4⋊C4—64no—
183C4⋊C4—121no—
184C22⋊C4—24no—
185C22⋊C4—34no—
186C22⋊C4—38no—
187C22⋊C4—38no—
188C22⋊C4—41no—
189C22⋊C4—64no—
190C22⋊C4—64no—
191C22⋊C4—64no—
192C42—64no—
193C42—121no—
194C22×C3—24no—
195C22×C3—24no—
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3C23×C411yescharacteristic
205C22×C3C22×D412yes—
206C22×C3C2×#4112yes—
207C22×C3H(5,𝔽2)11yescharacteristic
208C22×C3H(5,𝔽2)11yescharacteristic
209C2×D3—24no—
210C2×D3—24no—
211C2×D3—24no—
212C2×D3—24no—
213C2×D3—24no—
214C2×D3—24no—
215C2×D3—416no—
216C2×D3C4×D418yescomplemented
217C3×C4—22no—
218C3×C4—24no—
219C3×C4—42no—
220Dic3—24no—
221Dic3—24no—
222Dic3—42no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—34no—
232C23—34no—
233C23—34no—
234C23—616no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23—124no—
240C23—124no—
241C23—124no—
242C23—128no—
243C23C2×C4×D312yes—
244C23C2×C4×D312yes—
245C23C23×D311yescharacteristic
246D4—44no—
247D4—616no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—34no—
256C2×C4—34no—
257C2×C4—44no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—128no—
266C2×C4—128no—
267C2×C4C2×C4×D312yes—
268C2×C4D3×D414yes—
269C2×C3—42no—
270C2×C3—42no—
271C2×C3C2×C4×D412yes—
272C2×C3#51811yescharacteristic
273C2×C3D4211yescharacteristic
274C2×C3#54612yes—
275C2×C3#55111yescharacteristic
276D3—28no—
277C22—22no—
278C22—22no—
279C22—24no—
280C22—24no—
281C22—38no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—64no—
289C22—64no—
290C22—64no—
291C22—64no—
292C22—64no—
293C22—64no—
294C22—1216no—
295C22C22×C4×D311yescharacteristic
296C22C2×D3×D412yes—
297C22#100411yescharacteristic
298C22#100411yescharacteristic
299C22D3×#4112yes—
300C4—22no—
301C4—24no—
302C4—42no—
303C4—64no—
304C4—64no—
305C4—122no—
306C3#235811yescharacteristic, complemented, Sylow
307C2—42no—
308C2—42no—
309C2—68no—
310C2#548611yescharacteristic
311C2C4×D3×D412yes—
312C2#559011yescharacteristic
313C2#561212yes—
314C2#565911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#386C2
socle, cosocleC23×C3C24
prime radicalC2×#937C2
hypercenter#386D3
hypocenterC3#2358
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×#67C2semidirect
3. C2×D3×C22⋊C4C2semidirect
4. C3×#386C2semidirect
5. #5221C2semidirect
6. #4956C2semidirect
7. #4945C2semidirect
8. #4668C2semidirect
9. #4666C2semidirect
13. C23×C3×C4C22semidirect
18. C2×C3×C22⋊C4C22semidirect
21. C2×#217C22semidirect
22. #937C4semidirect
27. C2×C4⋊Dic3C22semidirect
28. C2×Dic3⋊C4C22semidirect
29. C2×C4×Dic3C22semidirect
30. C2×C4×Dic3C22semidirect
32. C3×#95C22semidirect
33. C3⋊#95C22semidirect
34. C3⋊#95C22semidirect
41. #386D3semidirect
50. C23×D3C2×C4semidirect
54. C2×C3×D4C2×C4semidirect
62. C2×#67C2×C4semidirect
76. #217C2×C4semidirect
78. (C2×D3)⋊C4C2×C4semidirect
79. (C2×D3)⋊C4D4semidirect
89. C23×C4C2×D3semidirect
102. C2×C22⋊C4C2×D3semidirect
107. #95C2×D3semidirect
167. C2×D4C4×D3semidirect
216. C2×D3C4×D4semidirect
306. C3#2358semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#9372
C2×C4×#672
C2×D3×C22⋊C42
C3×#3862
#52212
#49562
#49452
#46682
#46662
#23583